Any help would be appreciated! Problem 4 Let (X, Y)~ N and Z = X1(XY >...
7. Let X1, X2, ... be an i.i.d. random variables. (a) Show that max(X1,... , X,n)/n >0 in probability if nP(Xn > n) -» 0. (b) Find a random variable Y satisfying nP(Y > n) ->0 and E(Y) = Oo
Please any help would be appreciated Assume the following theorem: If R.V.'s X1, X2, ..., Xn are independent and uniformly bounded (i.e. JM > 0) such that the P(|X1| > M) = 0 and limn+_ V(Yn) = limn+oV(S1_, Xk) = ), then the distribution of the standardized mean of X; approaches the stan dard normal distribution. Now, consider the sequence of independent random variables (Xk) =1, and assume each has uniform density 1 0 < xk < fk(xk) = {...
6. Let X1, X2,.. , Xn denote a random sample of size n> 1 from a distribution with pdf f(x; 6) = 6e-8, 0<x< 20, zero elsewhere, and 0 > 0. Le Y = x. (a) Show that Y is a sufficient and complete statistics for . (b) Prove that (n-1)/Y is an unbiased estimator of 0.
5. Let F(x, y, z) = (yz, xz, xy) and define Cr,h = {(x, y, z) : x2 + y2 = p2, z = h}. 1 Show that for any r > 0 and h ER, Sony F. dx = 0
1. Let F(x, y, z) = (-y + ,2-2,2-y), and let S be the surface of the paraboloid 2 = 9-32 - v2 for 2 > 0. oriented by an upward pointing normal vector. Note that the boundary of S is C, the circle of radius 3 in the xy-plane. Verify Stokes' Theorem by computing both sides of the equality: (a) (1 Credit) || (D x F). ds (b) (1 Credit) $F. dr
Let z=5 where x, y, z E R. Prove that z? +z2+z?>
5. Let F(x, y, z) = (yz, xz, xy) and define 2 Crin = {(x,y,z) : x2 + y2 = r2, 2 = h} Show that for any r > 0 and h ER, le F. dx = 0 Crih
4. Let X, Y, and Z be independent random variables, each with the standard normal distribution. Compute the following: (a) P[X + Y> Z +2 (b) Var3x 4Y;
Let X1, X2,..., X, be n independent random variables sharing the same probability distribution with mean y and variance o? (> 1). Then, as n tends to infinity the distribution of the following random variable X1 + X2 + ... + x, nu vno converges to Select one: A. an exponential distribution B. a normal distribution with parameters hi and o? C a normal distribution with parameters 0 and 1 D. a Poisson distribution
7. Let X1,... , Xn be iid based on f(x; 6) -22e-z?/e where x > 0. Show that θ=-yx? is efficient